MAT 203 Final

American University College of Science and Technology MAT 203 FINAL EXAM Summer 2005/Tutorial Duration: 2.0 hours NAME: Instructions: The use of calculators is allowed. Kindly write your solutions ONLY on this exam booklet and show all work. No scratch paper is allowed. Make sure all exam pages are included. 1. (20 pts – 4 pts each) Determine whether the following series converges absolutely or conditionally or diverges. Justify your answer. a) 1 of 12 b) 2 of 12 c) 3 of 12 d) 4 of 12 e) 5 of 12 2. (10 pts) Use series to estimate the following integral with an error of magnitude less than 0.01. 6 of 12 3. (10 pts) Find the shared area between the circle r=2cosθ and the cardioid r=1+sinθ. 7 of 12 4. (15 pts) Use spherical coordinates to find the volume of the solid bounded below by the hemisphere ρ=1, z≥0, and above by the cardioid of revolution ρ =1+cosφ. 8 of 12 5. (10 pts) Use double integrals to find the volume of the region bounded above by the paraboloid z = 9 – x2 – y2 , below by the xy-plane , and lying outside the cylinder x2 + y2 =1 9 of 12 6. (10 pts) Change the Cartesian integral into an equivalent polar integral, then evaluate. 10 of 12 7. (15 pts) Using triple integrals to find the volume of the region bounded above by the paraboloid and below by the paraboloid . (Find only limits of integration.) 11 of 12 8. (10 points) Find the Fourier series of f(x) on the given interval. 12 of 12